Monotonicity: Difference between revisions
Added information about monotonicity/LNH/mutual majority incompatibility.
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In elections via the single-winner methods [[range voting]] and [[majority judgment]] nobody can help a candidate by reducing or removing support for them. The definition of the monotonicity criterion with regard to these methods is disputed. Some voting theorists argue that this means these methods pass the monotonicity criterion; others say that, as these are not ''ranked'' voting systems, they are out of the monotonicity criterion's scope.
=== Implications ===
It's impossible for a method to pass all of monotonicity, [[later-no-harm]], [[later-no-help]], and [[mutual majority]],<ref name="Woodall-Monotonicity"/> but there do exist methods that pass three of the four. [[First past the post]] passes the first three, [[instant-runoff voting]] passes the last three, and [[Descending Acquiescing Coalitions]] and [[Descending Solid Coalitions]] pass one of the later-no-help/harm criteria as well as monotonicity and mutual majority.
== Definition of monotonicity criteria==
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[[Douglas Woodall]] defined several variations or types of monotonicity.<ref name="Woodall-Monotonicity" /> In Woodall's definitions, a candidate x should not be harmed if:
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=== Multi-winner monotonicity ===
Monotonicity would be more aptly named ''endorsement monotonicity'' since it is the preservation of monotonicity relative to endorsement. Since it is the most important form of monotonicity is bears the simple naming. There are however two other important forms of monotonicity for multi-winner voting systems, [[Population monotonicity]] and [[House monotonicity criterion |House monotonicity]].
Multi-winner monotonicity could also be considered in a weaker and stronger sense: the weak form is satisfied whenever, if A is one of the winners, ranking A higher does not kick A out of the winning set; whereas the stronger form is satisfied whenever, if A is one of the winners, ranking A higher does not kick ''anyone'' out of the winning set. In a single winner election, these criteria are the same, but the stronger form is harder to satisfy for multi-winner. Woodall's definition of mono-raise corresponds to the weak form.
==Footnotes==
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[[Category:Voting system criteria]]
[[Category:Monotonic electoral systems|*]]
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